Counting connected components of a semialgebraic set in subexponential time

Let a semialgebraic set be given by a quantifier-free formula of the first-order theory of real closed fields withk atomic subformulae of the typef _i0 for 1ik, where the polynomialsf _i[X ₁,...,X _n] have degrees deg(f _i)<d and the absolute value of each (integer) coefficient off _i is at most 2M. An algorithm is exhibited which counts the number of connected components of the semialgebraic set in time (M (kd)^{n 20})^{O (1)}. Moreover, the algorithm allows us to determine whether any pair of points from the set are situated in the same connected component.     

Best Huffman trees

Summary Given a sequence of positive weights, W=w ₁...w _n >0, there is a Huffman tree, T (T-up) which minimizes the following functions: max{d(w_i)}; d(w_i); f(d(w_i)) w _i(here d(w _i) represents the distance of a leaf of weight w _i to the root and f is a function defined for nonnegative integers having the property that g(x) = f(x + 1) – f(x) is monotone increasing) over the set of all trees for W having minimal expected length. Minimizing the first two functions was first done by Schwartz [5]. In the case of codes where W is a sequence of probabilities, this implies that the codes based on T have all their absolute central moments minimal. In particular, they are the least variance codes which were also described by Kou [3]. Furthermore, there exists a Huffman tree T, (T-down) which maximizes the functions considered above.However, if g(x) is monotone decreasing, T and T, respectively maximize and minimize f(d(w_i) w _i) over the set of all trees for W having minimal expected length. In addition, we derive a number of interesting results about the distribution of labels within Huffman trees. By suitable modifications of the usual Huffman tree construction, (see [1]) T and T can also be constructed in time O(n log n).     

Three-valued representative systems

A representative system defined onn voters or propositionsi = 1,,n is a functionF: {1,0, -1} ⁿ {1,0, -1} which is monotonic (D E F(D) F(E)), unanimous (F(1,, 1) = 1), dual (F(-D) = -F(D)), and satisfies a positivity property which says that the set of all non-zero vectors in {1, 0, -1} ⁿ for whichF(D) = 0 can be partitioned into two dual subsets each of which has the property that ifD andE are in the subset thenD _i+E _i > 0 for somei. Representative systems can be defined recursively from the coordinate projectionsS _i (D) = D _i using sign functions, and in this format they are interpreted as hierarchical voting systems in which outcomes of votes in lower levels act as votes in higher levels of the system. For each positive integern, (n) is defined as the smallest positive integer such that all representative systems defined on {1, 0, -1} ⁿ can be characterized by(n) or fewer hierarchical levels. The function is nondecreasing inn, unbounded above, and satisfies(n) n–1 for alln. In addition,(n) = n–1 forn {1, 2, 3, 4}, and it is conjectured that does not continue to grow linearly asn increases.     

Sufficiency conditions for existence of an optimal feedback control in stochastic mechanics

In the framework of stochastic mechanics, the following problem is considered: in a set of admissible feedback controls v, with range inE ⁿ , find one minimizing the expectationE _sx { _s ^T L(t, (t), (t, (t)))dt + W _T ((T))} for all (s, x) [0,T) E ⁿ , whereL(t, x, ) = (/12)m ² – U(t, x) is the classical action integrand and is an-dimensional diffusion process in the weak sense, (see Bensoussan, 1982) with drift and diffusion coefficientD constant > 0.W _T andU are given real functions. Sufficiency conditions for the existence of such an optimal feedback control are given. Dedicated to George Leitmann Recommended by G.J. Olsder Presented at the Third Workshop on Control Mechanics in honor of George Leitmann, January 22–24, 1990, University of Southern California, Los Angeles, California (USA).     

Unextendible Product Bases and Locally Unconvertible Bound Entangled States

Mutual convertibility of bound entangled states under local quantum operations and classical communication (LOCC) is studied. We focus on states associated with unextendible product bases (UPB) in a system of three qubits. A complete classification of such UPBs is suggested. We prove that for any pair of UPBs S and T the associated bound entangled states _S and _T cannot be converted to each other by LOCC, unless S and T coincide up to local unitaries. More specifically, there exists a finite precision (S,T) > 0 such that for any LOCC protocol mapping _S into a probabilistic ensemble (p, ), the fidelity between _T and any possible final state satisfies F(_T, ) = 1 - (S,T).PACS: 03.65.Bz; 03.67.-a; 89.70+c.     

Systems of Strings with High Mutual Complexity

Consider a binary string x ₀ of Kolmogorov complexity K(x ₀) n. The question is whether there exist two strings x ₁ and x ₂ such that the approximate equalities K(x _i x _j ) n and K(x _i x _j , x _k ) n hold for all 0 i, j, k 2, i j k, i k. We prove that the answer is positive if we require the equalities to hold up to an additive term O(log K(x ₀)). It becomes negative in the case of better accuracy, namely, O(log n).     

The conjoinability relation in Lambek calculus and linear logic

In 1958 J. Lambek introduced a calculusL of syntactic types and defined an equivalence relation on types: x y means that there exists a sequence x=x₁,...,x_n=y (n 1), such thatx _i x _i+1 or x_i+ x _i (1 i n). He pointed out thatx y if and only if there is joinz such thatx z andy z. This paper gives an effective characterization of this equivalence for the Lambeck calculiL andLP, and for the multiplicative fragments of Girard's and Yetter's linear logics. Moreover, for the non-directed Lambek calculusLP and the multiplicative fragment of Girard's linear logic, we present linear time algorithms deciding whether two types are equal, and finding a join for them if they are.The author was sponsored by project NF 102/62-356 (Structural and Semantic Parallels in Natural Languages and Programming Languages), funded by the Netherlands Organization for the Advancement of Research (N.W.O.).     

Undirecteds-t connectivity in polynomial time and sublinear space

Thes-t connectivity problem for undirected graphs is to decide whether two designated vertices,s andt, are in the same connected component. This paper presents the first known deterministic algorithms solving undirecteds-t connectivity using sublinear space and polynomial time. Our algorithms provide a nearly smooth time-space tradeoff between depth-first search and Savitch's algorithm. Forn vertex,m edge graphs, the simplest of our algorithms uses spaceO(s),n ^1/2log₂ nsnlog₂ n, and timeO(((m+n)n ² log² n)/s). We give a variant of this method that is faster at the higher end of the space spectrum. For example, with space (nlogn), its time bound isO((m+n)logn), close to the optimal time for the problem. Another generalization uses less space, but more time: spaceO(n ^1/logn), for 2log₂ n, and timen ^O(). For constant the time remains polynomial.     

A Note on Scheduling Tall/Small Multiprocessor Tasks with Unit Processing Time to Minimize Maximum Tardiness

We study the scheduling situation where n tasks, subjected to release dates and due dates, have to be scheduled on m parallel processors. We show that, when tasks have unit processing times and either require 1 or m processors simultaneously, the minimum maximal tardiness can be computed in polynomial time. Two algorithms are described. The first one is based on a linear programming formulation of the problem while the second one is a combinatorial algorithm. The complexity status of this tall/small task scheduling problem P|r _i ,p _i =1, size _i {1,m}|T _max was unknown before, even for two processors.     

Shallow circuits and concise formulae for multiple addition and multiplication

A theory is developed for the construction of carry-save networks with minimal delay, using a given collection of carry-save adders each of which may receive inputs and produce outputs using several different representation standards.The construction of some new carry-save adders is described. Using these carry-save adders optimally, as prescribed by the above theory, we get {, , }-circuits of depth 3.48 log₂ n and {, , }-circuits of depth 4.95 log₂ n for the carry-save addition ofn numbers of arbitrary length. As a consequence we get multiplication circuits of the same depth. These circuits put out two numbers whose sum is the result of the multiplication. If a single output number is required then the depth of the multiplication circuits increases respectively to 4.48 log₂ n and 5.95 log₂ n.We also get {, , }-formulae of sizeO (n ^3.13) and {, }-formulae of sizeO (n ^4.57) for all the output bits of a carry-save addition ofn numbers. As a consequence we get formulae of the same size for the majority function and many other symmetric Boolean functions.     

Hierarchy theorems for two-way finite state transducers

Summary For a family of languages , CAL() is defined as the family of images of under nondeterministic two-way finite state transducers, while FINITE · VISIT() is the closure of under deterministic two-way finite state transducers; CAL⁰()= and for n0, CAL ⁿ⁺¹()=CAL ⁿ (CAL()). For any semiAFL , if FINITE · VISIT() CAL(), then CAL ⁿ () forms a proper hierarchy and for every n0, FINITE · VISIT(CALⁿ()) CAL ⁿ⁺¹() FINITE · VISIT(CAL ⁿ⁺¹()). If is a SLIP semiAFL or a weakly k-iterative full semiAFL or a semiAFL contained in any full bounded AFL, then FINITE · VISIT() CAL() and in the last two cases, FINITE · VISIT(). If is a substitution closed full principal semiAFL and FINITE · VISIT(), then FINITE · VISIT() CAL(). If is a substitution closed full principal semiAFL generated by a language without an infinite regular set and ₁ is a full semiAFL, then is contained in CAL^m(₁) if and only if it is contained in ₁. Among the applications of these results are the following. For the following families , CAL ⁿ () forms a proper hierarchy: =INDEXED, =ETOL, and any semiAFL contained in CF. The family CF is incomparable with CAL ^m (NESA) where NESA is the family of one-way nonerasing stack languages and INDEXED is incomparable with CAL ^m (STACK) where STACK is the family of one-way stack languages.This work was supported in part by the National Science Foundation under Grants No. DCR74-15091 and MCS-78-04725     

Separating complexity classes related to bounded alternating ω-branching programs

We develop a theory of communication within branching programs that provides exponential lower bounds on the size of branching programs that are bounded alternating. Our theory is based on the algebraic concept of -branching programs, : , a semiring homomorphism, that generalizes ordinary branching programs, -branching programs [M2] andMOD _p-branching programs [DKMW].Due to certain exponential lower and polynomial upper bounds on the size of bounded alternating -branching programs we are able to separate the corresponding complexity classesN _ba ,co-N _{ba ba} , andMOD _p - _ba ,p prime, from each other, and from that classes corresponding to oblivious linear length-bounded branching programs investigated in the past.     

Duality in Quantum Information Geometry

Let H be a separable Hilbert space. We consider the manifold M consisting of density operators on H such that ^p is of trace class for some p (0, 1). We say M is nearby if there exists C > 1 such that C ^–1C. We show that the space of nearby points to can be furnished with the two flat connections known as the (±)-affine structures, which are dual relative to the BKM metric. We furnish M with a norm making it into a Banach manifold.     

On the group of automorphisms of strongly connected automata

LetA = (S, I, M) be a strongly connected finite automaton withn states. Weeg has shown that ifA has a group of automorphisms of orderm, then there is a partition of the setS inton/m blocks each withm states. Furthermore ifs _i ands _j are in the same block of, thenT _ii =T _jj , whereT _ii = {x|x ^* and thenM(s _i , x) =s _i }. It will be shown that the partition also must have the substitution property and that these two conditions are sufficient for ann state strongly connected automaton to have a group of automorphisms of orderm.Necessary and sufficient conditions for twon-state strongly connected automata to have isomorphic automorphism groups are given. Also, it is demonstrated that forT _ii to equalT _jj it is necessary to check only a finite number of tapes and consequently provide an algorithm for determining whether or notA has a group of automorphisms of orderm. This research was partially supported by the National Science Foundation under Grant No. G.P. 7077.     

Gradually intractable problems and nondeterministic log-space lower bounds

The paper places five different problems (thek-pebble game problem, two problems aboutk finite automata, the reachability problem for Petri nets withk tokens, and the teachability problem for graphs whose k-dimensional edge sets are described by Cartesian products ofk factors) into the hierarchyNL _k of problems solvable by nondeterministic Turing machines ink-log₂ n space (and binary tape alphabet, to avoid tape speed-up). The results, when combined with the conjecture thatNL _i contains problems that requireO(n ^k ) deterministic time, show that these problems, while inP for every fixed value ofk, have polynomial deterministic time complexities with the degree of the polynomial growing linearly with the parameterk, and hence are, in this sense, gradually intractable.     

Directional differentiation of marginal functions in mathematical programming problems

We consider regular mathematical programming problems of the form f(x, y) inf, y F(x), x Rⁿ, where F(x) = {y R^m h_i| (x, y) 0, , h_i (x, y) = 0, . The directional derivatives offunctions (x) = inf{f(x, y)|y F(x)} are estimated.Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 70–77, November–December, 1991.     

A forbidden subposet characterization of an order — dimension inequality

Dushnik and Miller defined the dimension of a partially ordered setX, DimX, as the minimum number of linear extensions ofX whose intersection is the partial ordering onX. The concept of dimension for a partially ordered set has applications to preference structures and the theory of measurement. Hiraguchi proved that DimX [|X|/2] when |X| 4. Bogart, Trotter, and Kimble gave a forbidden subposet characterization of Hiraguchi's inequality by constructing for eachn 2 the minimum collection _n of posets such that if [|X|/2] =n 2, then DimX < n unlessX contains one of the posets in _n . Recently Trotter gave a simple proof of Hiraguchi's inequality based on the following theorem. IfA is an antichain ofX and |X – A| =n 2, then DimX n. In this paper we give a forbidden subposet characterization of this last inequality.     

Variable Precision Algorithm for the Numerical Computation of the Fermi–Dirac Function ℱj(x) of Order j=−3/2

The purpose of this technical note is to present a piecewise Chebyshev expansion for the numerical computation of the Fermi–Dirac function _–3/2(x), –<x<. The variable precision algorithm we given automatically adjusts the degrees of the Chebyshev expansions so that _–3/2(x) can be efficiently computed to d significant decimal digits of accuracy, for a user specified value of d in the range 1d15.     

Polynomial-Time Decomposition Algorithms for Support Vector Machines

This paper studies the convergence properties of a general class of decomposition algorithms for support vector machines (SVMs). We provide a model algorithm for decomposition, and prove necessary and sufficient conditions for stepwise improvement of this algorithm. We introduce a simple rate certifying condition and prove a polynomial-time bound on the rate of convergence of the model algorithm when it satisfies this condition. Although it is not clear that existing SVM algorithms satisfy this condition, we provide a version of the model algorithm that does. For this algorithm we show that when the slack multiplier C satisfies 1/2 C mL, where m is the number of samples and L is a matrix norm, then it takes no more than 4LC ² m ⁴/ iterations to drive the criterion to within of its optimum.     

Splitting a configuration in a simplex

This paper presents a new method of partition, named-splitting, of a point set ind-dimensional space. Given a pointG in ad-dimensional simplexT, T(G;i) is the subsimplex spanned by G and the ith facet ofT. LetS be a set ofn points inT, and let be a sequence of nonnegative integers ₁, ..., n_d+1 satisfying _i=1 ^d+1 ₁=n The-splitter of (T, S) is a pointG inT such thatT(G;i) contains at least _i points ofS in its closure for everyi=1, 2, ...,d + 1. The associated dissection is the re-splitting.The existence of a-splitting is shown for any (T, S) and, and two efficient algorithms for finding such a splitting are given. One runs inO(d²n logn + d³n) time, and the other runs inO(n) time if the dimensiond can be considered as a constant. Applications of re-splitting to mesh generation, polygonal-tour generation, and a combinatorial assignment problem are given.